Questions: Describe the sampling distribution of p̂. Assume the size of the population is 25,000. n=800, p=0.1 Choose the phrase that best describes the shape of the sampling distribution of p̂ below. A. Approximately normal because n ≤ 0.05 N and np(1-p) ≥ 10. B. Not normal because n ≤ 0.05 N and np(1-p) < 10. C. Not normal because n ≤ 0.05 N and np(1-p) ≥ 10. D. Approximately normal because n ≤ 0.05 N and np(1-p) < 10.

Describe the sampling distribution of p̂. Assume the size of the population is 25,000.
n=800, p=0.1

Choose the phrase that best describes the shape of the sampling distribution of p̂ below.

A. Approximately normal because n ≤ 0.05 N and np(1-p) ≥ 10.
B. Not normal because n ≤ 0.05 N and np(1-p) < 10.
C. Not normal because n ≤ 0.05 N and np(1-p) ≥ 10.
D. Approximately normal because n ≤ 0.05 N and np(1-p) < 10.

Transcript text: Describe the sampling distribution of $\hat{p}$. Assume the size of the population is 25,000 . \[ n=800, p=0.1 \] Choose the phrase that best describes the shape of the sampling distribution of $\hat{p}$ below. A. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. B. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. C. Not normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. D. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})<10$.

Solution

Solution Steps

Step 1: Determining the Shape of the Sampling Distribution

To determine if the sampling distribution of $\hat{p}$ is approximately normal, we check two conditions:

The sample size $n$ is less than or equal to 5% of the population size $N$: $n \leq 0.05N$ is met.
The product $np(1-p)$ is greater than or equal to 10: $np(1-p) \geq 10$ is met. Since both conditions are met, the sampling distribution of $\hat{p}$ is approximately normal.

Step 2: Calculating the Mean of the Sampling Distribution

The mean of the sampling distribution of $\hat{p}$, denoted as $\mu_{\hat{p}}$, is equal to the population proportion $p$: $\mu_{\hat{p}} = 0.1$.

Step 3: Calculating the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of $\hat{p}$, denoted as $\sigma_{\hat{p}}$, is calculated using the formula $\sqrt{\frac{p(1-p)}{n}}$: $\sigma_{\hat{p}} = 0.01$.

Final Answer:

The sampling distribution of $\hat{p}$ is approximately normal with a mean of $\mu_{\hat{p}} = 0.1$ and a standard deviation of $\sigma_{\hat{p}} = 0.01$.

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